Quantum matter is weakly entangled at low energies

We construct upper bounds on entanglement entropies of many-body quantum states that have fixed energy expectation values with respect to local Hamiltonians. Our focus is on entanglement entropies of subsystems that make up approximately half of the full system. The upper bound on the von Neumann entanglement entropy is half the sum of the thermal entropies of two fictitious systems at the same temperature as one another, with an additional area-law contribution in some systems. The effective temperature is chosen such that the sum of the thermal energies of the two fictitious systems matches the constraint on the energy of the state in the original problem; at subextensive energies, this temperature decreases with increasing system size. Our upper bounds on Rényi entanglement entropies take an analogous form. These inequalities show that, in systems that are frustration-free (FF) and whose zero-temperature subsystem thermal entropies scale with surface areas (rather than subsystem volumes), ground states are area-law entangled. For physical models of quantum matter, which have well-defined specific heat capacities (and are not necessarily FF), our bounds provide a way to convert this thermodynamic data into constraints on pure-state entanglement at both subextensive and extensive energies. We also show that our general upper bounds on half-system entanglement entropies are optimal, up to subleading corrections, in systems with reflection symmetry and above a subextensive energy threshold. A key application of our results is to constrain the entanglement of quantum states relevant to simulations of zero-temperature response functions, which have energies of order unity. We prove that, in wide varieties of FF systems in two dimensions and higher, the contributions of excitations to von Neumann entanglement entropy are subleading compared with the contribution from the ground state.